Optimization of Economic Production Quantity and Profits under Markovian Demand

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1 From the SelectedWorks of Innovative Research Publications IRP India Winter January 1, 2015 Optimization of Economic Production Quantity Profits under Markovian Dem Innovative Research Publications, IRP India, Innovative Research Publications Kizito Paul Mubiru Available at:

2 Optimization of Economic Production Quantity Profits under Markovian Dem Kizito Paul Mubiru, Kyambogo University Abstract--- In most manufacturing industries; dem uncertainty affects effective planning control of production levels that maximize profits. In this paper, a new mathematical model is developed to optimize economic production quantity (EPQ) of a single-item, finite horizon, periodic review inventory problem with Markovian dem. In this model, sales price inventory replenishment periods are uniformly fixed over the planning horizon. Adopting a Markov decision process approach, the states of a Markov chain represent possible states of dem for the inventory item. The production cost, holding cost, shortage cost sales price are combined with dem inventory positions to generate the profit for the decision problem. The objective is to determine in each period of the planning horizon an optimal economic production quantity so that the long run profits are maximized for a given state of dem. Using weekly equal intervals, the decisions of how much to produce are made using dynamic programming over a finite period planning horizon. A numerical example demonstrates the existence of an optimal state-dependent economic production quantity as well as the corresponding profits of item. Keywords- Economic Production Quantity, Markovian dem, Profits 1. INTRODUCTION In production-inventory management, the zeal for manufacturing industries to plan for optimal production levels that sustain rom dem leaves a lot to be desired. In practice, when production exceeds quantity demed, inventory carrying costs accumulate which affect profit margins of the manufacturer. Similarly, production levels below dem impose shortage costs loss of good will from potential customers. Both cases drastically reduce profit margins unless proper planning coordination is put in place to establish optimal production levels in a given manufacturing industry. In an effort to achieve this goal, two major problems are usually encountered: (i)determining the economic production quantity (EPQ) of the item in question (ii)determining the optimal profits associated with the economic production quantity(epq) when dem is uncertain. In this paper, a production-inventory system is considered whose goal is to optimize the economic production quantity (EPQ) profits associated with producing holding inventory of an item. At the beginning of each period, a major has to be made: namely whether to produce additional units of the item in inventory or cancel production utilize the available units in inventory. In either case, the economic production quantity (EPQ) must be determined that optimize profits in a given production-inventory system. According to Goya[1],the EPQ may be computed each time a product is scheduled for production when all products are produced on a single machine. The distribution of lot size can be computed when the dem for each product is a stochastic variable with a known distribution. The distribution is independent for non-overlapping time periods identical for equal time epochs. In related work by Khouja[2],the EPQ is shown to be determined under conditions of increasing dem, shortages partial backlogging. In this case the unit production cost is a function of the production process the quality of the production process deteriorates with increased production rate.tabucanno Mario [3] also presented a production order quantity model with stochastic dem for a chocolate milk manufacturer. In this paper, production planning is made complex by the stochastic nature of dem for the manufacturer s product. An analysis of production planning via dynamic programming approach shows that the company s production order variable is convex. In the article presented by Kampf Kochel[4], the stochastic lot sizing problem is examined where the cost of waiting lost dem is taken into consideration. In this model, production planning is made possible by making a trade-off between the cost of waiting the lost dem. The four models presented have some interesting insights regarding economic production quantity in terms of optimization stochastic dem. However, profit maximization is not a salient factor in the optimization models cited. 2. MODEL FORMULATION 2.1 Notation Assumptions i,j = States of dem N Z ij = Number of customers D Z ij = Quantity demed F = Favorable state U = Unfavorable state I Z ij = Quantity in inventory n,n = Stages P Z ij = Profits Z = Production lot sizing policy V Z i = Expected profits N Z = Customer matrix a Z i = Accumulated profits D Z = Dem matrix c p = Unit production cost I Z = Inventory matrix IJER@2015 Page 18

3 c h = Unit holding cost P Z = Profit matrix c s = Unit shortage cost Q Z =Dem transition matrix Q Z ij= Dem transition probability p s = Unit sales price E Z i = Economic Production Quantity i,j є {F,U} Z є {0,1} n=1,2,.n We consider a production-inventory system of a single item whose dem during a chosen period over a fixed planning horizon is classified as either Favorable (state F) or Unfavorable (state U) the dem of any such period is assumed to depend on the dem of the preceding period. The transition probabilities over the planning horizon from one dem state to another may be described by means on a Markov chain. Suppose one is interested in determining the optimal course of action, namely to produce additional units of item (a decision denoted by Z=1) or not to produce additional units (a decision denoted by Z=0) during each time period over the planning horizon where Z is a binary decision variable. Optimality is defined such that the maximum expected profits are accumulated at the end of N consecutive time periods spanning the planning horizon. In this paper, a two-period (N=2) planning horizon is considered. 2.2 Finite dynamic programming formulation Recalling that the dem can be in state F or in state U, the problem of finding an optimal economic production quantity may be expressed as a finite period dynamic programming model. Let g n (i) denote the expected total profits accumulated during periods n,n+1,...,n given that the state of the system at the beginning of period n is iε{f,u}.the recursive equation relation g n g n+1 is iε{f,u},n=1,2,...n conditions (1) together with the final This recursive relationship may be justified by noting that the cumulative total profits P Z ij+ g n+1 (j) resulting from reaching jε{f,u }at the start of period n+1 from state iε{f,u}at the start of period n occurs with probability Q Z ij. Clearly, V Z = Q Z (P Z ) T, Zε {0,1} (2) where T denotes matrix transposition, hence the dynamic programming recursive equations iε{f,u} n=1,2,...n-1, Zε{0,1}. iε{f,u} (4) result where (4) represents the Markov chain stable state. (3) The dem transition probability from state i ε{f, U}to state jε{f, U } given production lot sizing policy Zε {0,1} may be taken as the number of customers observed when dem is initially in state i later with dem changing to state j, divided by the number of customers over all states. (5) iε{f,u}, Zε {0,1} When dem outweighs on-h inventory, the profit matrix P Z may be computed by the relation: Therefore for all i,j ε{f,u}, Zε {0,1} A justification for expression (6) is that D Z ij -I Z ij units must be produced in order to meet the excess dem. Otherwise production is cancelled when dem is less than or equal to on-h inventory. The economic production quantity when dem is initially in state i ε{f,u},given production lot sizing policy Zε {0,1} is iε{f,u}, Zε {0,1} (7) The following conditions must however hold: 1. E Z i > 0 when D Z ij > I Z ij D Z ij = 0 when D Z ij I Z ij 2. Z=1 when c p > 0, Z=0 when c p = 0 3. c s > 0 when shortages are allowed c s = 0 when shortages are not allowed. 4. COMPUTING AN OPTIMAL ECONOMIC PRODUCTION QUANTITY The optimal EPQ is found in this section for each time period separately, 4.1 Optimization during period 1 When dem is Favourable (i.e. in state F), the optimal production lot sizing decision during period 1 is The associated total profits EPQ are then respectively. Similarly, when dem is Unfavorable (i.e. in state U), the optimal production lot sizing policy during period 1 is (6) 3. COMPUTING Q Z, P Z E Z IJER@2015 Page 19

4 In this case, the associated total profits EPQ are 5.1 Data Collection A sample of 60 customers was used. Past data revealed the following dem pattern inventory levels over the state transitions for twelve weeks in Tables 1 2 below: Table 1 respectively. 4.2 Optimization during period 2 Using dynamic programming recursive equation (1), recalling that a Z i denotes the already accumulated profits at the end of period 1 as a result of decisions made during that period,it follows that: Therefore when dem is favorable (i.e. in state F), the optimal production lot sizing decision during period 2 is while the associated total profits EPQ are Similarly, when dem is Unfavorable (i.e. in state U), the optimal production lot sizing policy during period 2 is Customers, Dem Inventory Levels at state-transitions over twelve weeks for Production Lot sizing Policy 1(Z = 1) State Transition Customers Dem Inventory (i,j) N 1 ij D 1 ij I 1 ij FF FU UF UU Table 2 Customers, Dem Inventory Levels at state-transitions over twelve weeks for Production Lot sizing Policy 0(Z=0) State Transition Customers Dem Inventory (i,j) N 0 ij D 0 ij I 0 ij FF FU UF UU When additional mattresses are produced (Z=1) during week 1, while if additional mattresses are not produced during week 1, these matrices are while the associated total profits EPQ are In either case, the unit sales price(p s ) is $20.00,the unit production cost (c p ) is $15.00,the unit holding cost per week(c h ) is $0.50, the unit shortage cost per week(c s ) is $ CASE STUDY In order to demonstrate the use of the model in 2-4, a real case application from Vita foam, a manufacturer of mattresses in Uga is presented in this section. The dem for mattresses fluctuates from month to month. The company wants to avoid mover-producing when dem is low or under-producing when dem is high, hence seeks decision support in terms of an optimal production lot sizing policy, the associated profits specifically, a recommendation as to the EPQ of mattresses over a two-week period. 5.2 Computation of model parameters Using (5) (6), the state transition matrix the profit matrix (in thous dollars) for week 1 are for the case where additional mattresses are produced during week 1, while these matrices are given by for the case when additional mattresses are not produced during week 1. IJER@2015 Page 20

5 When additional mattresses are produced (Z=1), the matrices Q 1 P 1 yield the profits (in thous dollars) However, when additional mattresses are not produced (Z=0), the matrices Q 0 P 0 yield the profits (in US dollars) Since > 0.786, it follows that Z=0 is an optimal production lot sizing policy for week 2 with associated accumulated profits of $1.209 an EPQ of 0 units for the case of unfavorable dem. When shortages are not allowed, the values of Z g n (i) E Z i may be computed for i ε{f, U}in a similar fashion after substituting c s = 0 the matrix function P S = p s [D Z ] (c p + c h + c s )[D Z I Z ] 5.2 The optimal production lot sizing policy EPQ Since > 0.408, it follows that Z=1 is an optimal production lot sizing policy for week 1 with associated total profits of $1.488 an EPQ of = 6 units when dem is favorable. Since > 0.177, it follows that Z = 0 is an optimal production lot sizing policy for week 1 with associated total profits of $0.430 an EPQ of 0 units when dem is unfavorable. If dem is favorable, then the accumulated profits at the end of week 1 are Since > 1.367, it follows that Z = 1 is an optimal production lot sizing policy for week 1 with associated accumulated profits of $ an EPQ of = 6 units for the case of favorable dem. However, if dem is unfavorable, then the accumulated profits at the end of week 1 are 6. CONCLUSION A production -inventory model was presented in this paper. The model determines an optimal production lot sizing policy, profits the EPQ of a given item with stochastic dem. The decision of whether or not to produce additional inventory units is modeled as a multi-period decision problem using dynamic programming over a finite period planning horizon. The working of the model was demonstrated by means of a real case study. Acknowledgment The author wishes to thank the staff management of Vita foam for all the assistance rendered during the data collection exercise. REFERENCES i. Goyal S, 1974, Lot size scheduling on a single machine for stochastic dem, Management science, 19(11), ii. Khouja M &Mehrez A, 1994, Economic Production lot size model with varying production rate imperfect quality, Journal of Operations Research Society, 45(2), iii. Tabuccano & Mario T, 1997, A Production order quantity model with stochastic dem for a chocolate milk manufacturer, Journal of ProductionEconomics,21(2), iv. Kampf M &Kochel P, 2004, Scheduling lot size optimization of a production inventory system with multiple items using simulation parallel genetic Algorithm, Operations Research Proceedings, Springer-Heidelberg. IJER@2015 Page 21